How to Win Roulette | Roulette System | Winning Roulette Strategy
roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38...
Transcript of roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38...
![Page 1: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/1.jpg)
Roulette
The game of roulette serves as a nice introduction to some of thekey ideas relating to probability.
We will look at some of thepossible outcomes when a roulette wheel is spun and relate thoseoutcomes to probability in general.
A roulette wheel contains 38 slots, numbered 0, 00, and1, 2, 3, . . . 36. When the wheel is spun, a ball eventually falls intoone of the slots. Assuming the wheel is balanced and the slots arethe same size, as is supposed to be the case, there are 38 possible
outcomes, each of probability1
38.
This is an example of an Equiprobable Probability Space.
Alan H. SteinUniversity of Connecticut
![Page 2: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/2.jpg)
Roulette
The game of roulette serves as a nice introduction to some of thekey ideas relating to probability. We will look at some of thepossible outcomes when a roulette wheel is spun and relate thoseoutcomes to probability in general.
A roulette wheel contains 38 slots, numbered 0, 00, and1, 2, 3, . . . 36. When the wheel is spun, a ball eventually falls intoone of the slots. Assuming the wheel is balanced and the slots arethe same size, as is supposed to be the case, there are 38 possible
outcomes, each of probability1
38.
This is an example of an Equiprobable Probability Space.
Alan H. SteinUniversity of Connecticut
![Page 3: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/3.jpg)
Roulette
The game of roulette serves as a nice introduction to some of thekey ideas relating to probability. We will look at some of thepossible outcomes when a roulette wheel is spun and relate thoseoutcomes to probability in general.
A roulette wheel contains 38 slots, numbered 0, 00, and1, 2, 3, . . . 36.
When the wheel is spun, a ball eventually falls intoone of the slots. Assuming the wheel is balanced and the slots arethe same size, as is supposed to be the case, there are 38 possible
outcomes, each of probability1
38.
This is an example of an Equiprobable Probability Space.
Alan H. SteinUniversity of Connecticut
![Page 4: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/4.jpg)
Roulette
The game of roulette serves as a nice introduction to some of thekey ideas relating to probability. We will look at some of thepossible outcomes when a roulette wheel is spun and relate thoseoutcomes to probability in general.
A roulette wheel contains 38 slots, numbered 0, 00, and1, 2, 3, . . . 36. When the wheel is spun, a ball eventually falls intoone of the slots.
Assuming the wheel is balanced and the slots arethe same size, as is supposed to be the case, there are 38 possible
outcomes, each of probability1
38.
This is an example of an Equiprobable Probability Space.
Alan H. SteinUniversity of Connecticut
![Page 5: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/5.jpg)
Roulette
The game of roulette serves as a nice introduction to some of thekey ideas relating to probability. We will look at some of thepossible outcomes when a roulette wheel is spun and relate thoseoutcomes to probability in general.
A roulette wheel contains 38 slots, numbered 0, 00, and1, 2, 3, . . . 36. When the wheel is spun, a ball eventually falls intoone of the slots. Assuming the wheel is balanced and the slots arethe same size, as is supposed to be the case, there are 38 possible
outcomes, each of probability1
38.
This is an example of an Equiprobable Probability Space.
Alan H. SteinUniversity of Connecticut
![Page 6: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/6.jpg)
Roulette
The game of roulette serves as a nice introduction to some of thekey ideas relating to probability. We will look at some of thepossible outcomes when a roulette wheel is spun and relate thoseoutcomes to probability in general.
A roulette wheel contains 38 slots, numbered 0, 00, and1, 2, 3, . . . 36. When the wheel is spun, a ball eventually falls intoone of the slots. Assuming the wheel is balanced and the slots arethe same size, as is supposed to be the case, there are 38 possible
outcomes, each of probability1
38.
This is an example of an Equiprobable Probability Space.
Alan H. SteinUniversity of Connecticut
![Page 7: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/7.jpg)
Terminology
We consider experiments with a finite set of possible outcomes.
We call the set of possible outcomes the sample space and theindividual outcomes are called sample points.
A subset of the sample space is referred to as an event. We’llspend time on events later on.
Alan H. SteinUniversity of Connecticut
![Page 8: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/8.jpg)
Terminology
We consider experiments with a finite set of possible outcomes.
We call the set of possible outcomes the sample space
and theindividual outcomes are called sample points.
A subset of the sample space is referred to as an event. We’llspend time on events later on.
Alan H. SteinUniversity of Connecticut
![Page 9: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/9.jpg)
Terminology
We consider experiments with a finite set of possible outcomes.
We call the set of possible outcomes the sample space and theindividual outcomes are called sample points.
A subset of the sample space is referred to as an event. We’llspend time on events later on.
Alan H. SteinUniversity of Connecticut
![Page 10: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/10.jpg)
Terminology
We consider experiments with a finite set of possible outcomes.
We call the set of possible outcomes the sample space and theindividual outcomes are called sample points.
A subset of the sample space is referred to as an event. We’llspend time on events later on.
Alan H. SteinUniversity of Connecticut
![Page 11: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/11.jpg)
Straight Bets
There is a variety of ways one can bet at roulette. We will see thecasino doesn’t care how one bets.
The simplest bet is a straightbet, where one bets on a specific number and wins, with a payoffat 35-1, if that number comes up. This means that if a player bets$1 and wins, the player will get $36 back, the dollar he bet alongwith $35 more.
In our analyses, let’s assume every bet is for $1.
Alan H. SteinUniversity of Connecticut
![Page 12: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/12.jpg)
Straight Bets
There is a variety of ways one can bet at roulette. We will see thecasino doesn’t care how one bets. The simplest bet is a straightbet, where one bets on a specific number and wins, with a payoffat 35-1, if that number comes up.
This means that if a player bets$1 and wins, the player will get $36 back, the dollar he bet alongwith $35 more.
In our analyses, let’s assume every bet is for $1.
Alan H. SteinUniversity of Connecticut
![Page 13: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/13.jpg)
Straight Bets
There is a variety of ways one can bet at roulette. We will see thecasino doesn’t care how one bets. The simplest bet is a straightbet, where one bets on a specific number and wins, with a payoffat 35-1, if that number comes up. This means that if a player bets$1 and wins, the player will get $36 back, the dollar he bet alongwith $35 more.
In our analyses, let’s assume every bet is for $1.
Alan H. SteinUniversity of Connecticut
![Page 14: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/14.jpg)
Straight Bets
There is a variety of ways one can bet at roulette. We will see thecasino doesn’t care how one bets. The simplest bet is a straightbet, where one bets on a specific number and wins, with a payoffat 35-1, if that number comes up. This means that if a player bets$1 and wins, the player will get $36 back, the dollar he bet alongwith $35 more.
In our analyses, let’s assume every bet is for $1.
Alan H. SteinUniversity of Connecticut
![Page 15: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/15.jpg)
Straight Bets – The Big Picture
Suppose someone repeatedly makes a straight bet for $1.Although the results will vary, on average
the player will win about1
38of the time and lose the rest of the time.
If he (or she) plays 38 times, he could expect to win about onetime, coming out $35 ahead that time, but losing about 37 times,coming out $1 behind those times, so after playing 38 times he canexpect to be about $2 behind. That works out to losing about
$2
38= $
1
19per bet.
Alan H. SteinUniversity of Connecticut
![Page 16: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/16.jpg)
Straight Bets – The Big Picture
Suppose someone repeatedly makes a straight bet for $1.Although the results will vary, on average the player will win about1
38of the time and lose the rest of the time.
If he (or she) plays 38 times, he could expect to win about onetime, coming out $35 ahead that time, but losing about 37 times,coming out $1 behind those times, so after playing 38 times he canexpect to be about $2 behind. That works out to losing about
$2
38= $
1
19per bet.
Alan H. SteinUniversity of Connecticut
![Page 17: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/17.jpg)
Straight Bets – The Big Picture
Suppose someone repeatedly makes a straight bet for $1.Although the results will vary, on average the player will win about1
38of the time and lose the rest of the time.
If he (or she) plays 38 times, he could expect to win about onetime, coming out $35 ahead that time,
but losing about 37 times,coming out $1 behind those times, so after playing 38 times he canexpect to be about $2 behind. That works out to losing about
$2
38= $
1
19per bet.
Alan H. SteinUniversity of Connecticut
![Page 18: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/18.jpg)
Straight Bets – The Big Picture
Suppose someone repeatedly makes a straight bet for $1.Although the results will vary, on average the player will win about1
38of the time and lose the rest of the time.
If he (or she) plays 38 times, he could expect to win about onetime, coming out $35 ahead that time, but losing about 37 times,coming out $1 behind those times,
so after playing 38 times he canexpect to be about $2 behind. That works out to losing about
$2
38= $
1
19per bet.
Alan H. SteinUniversity of Connecticut
![Page 19: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/19.jpg)
Straight Bets – The Big Picture
Suppose someone repeatedly makes a straight bet for $1.Although the results will vary, on average the player will win about1
38of the time and lose the rest of the time.
If he (or she) plays 38 times, he could expect to win about onetime, coming out $35 ahead that time, but losing about 37 times,coming out $1 behind those times, so after playing 38 times he canexpect to be about $2 behind.
That works out to losing about
$2
38= $
1
19per bet.
Alan H. SteinUniversity of Connecticut
![Page 20: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/20.jpg)
Straight Bets – The Big Picture
Suppose someone repeatedly makes a straight bet for $1.Although the results will vary, on average the player will win about1
38of the time and lose the rest of the time.
If he (or she) plays 38 times, he could expect to win about onetime, coming out $35 ahead that time, but losing about 37 times,coming out $1 behind those times, so after playing 38 times he canexpect to be about $2 behind. That works out to losing about
$2
38= $
1
19per bet.
Alan H. SteinUniversity of Connecticut
![Page 21: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/21.jpg)
Random Variables
Random variables are numerical values associated with outcomesof experiments.
The letter X is the most common symbol used to represent arandom variable.
If we consider a straight bet of $1 to be an experiment and X tobe the net winnings for the player, then X can take on either thevalue of 35 (if the player wins) or −1 (if the player loses).
If we denote the probability X takes on a certain value k by
P(X = k), then P(X = 35) =1
38and P(X = −1) =
37
38.
Alan H. SteinUniversity of Connecticut
![Page 22: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/22.jpg)
Random Variables
Random variables are numerical values associated with outcomesof experiments.
The letter X is the most common symbol used to represent arandom variable.
If we consider a straight bet of $1 to be an experiment and X tobe the net winnings for the player, then X can take on either thevalue of 35 (if the player wins) or −1 (if the player loses).
If we denote the probability X takes on a certain value k by
P(X = k), then P(X = 35) =1
38and P(X = −1) =
37
38.
Alan H. SteinUniversity of Connecticut
![Page 23: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/23.jpg)
Random Variables
Random variables are numerical values associated with outcomesof experiments.
The letter X is the most common symbol used to represent arandom variable.
If we consider a straight bet of $1 to be an experiment and X tobe the net winnings for the player,
then X can take on either thevalue of 35 (if the player wins) or −1 (if the player loses).
If we denote the probability X takes on a certain value k by
P(X = k), then P(X = 35) =1
38and P(X = −1) =
37
38.
Alan H. SteinUniversity of Connecticut
![Page 24: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/24.jpg)
Random Variables
Random variables are numerical values associated with outcomesof experiments.
The letter X is the most common symbol used to represent arandom variable.
If we consider a straight bet of $1 to be an experiment and X tobe the net winnings for the player, then X can take on either thevalue of 35 (if the player wins)
or −1 (if the player loses).
If we denote the probability X takes on a certain value k by
P(X = k), then P(X = 35) =1
38and P(X = −1) =
37
38.
Alan H. SteinUniversity of Connecticut
![Page 25: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/25.jpg)
Random Variables
Random variables are numerical values associated with outcomesof experiments.
The letter X is the most common symbol used to represent arandom variable.
If we consider a straight bet of $1 to be an experiment and X tobe the net winnings for the player, then X can take on either thevalue of 35 (if the player wins) or −1 (if the player loses).
If we denote the probability X takes on a certain value k by
P(X = k), then P(X = 35) =1
38and P(X = −1) =
37
38.
Alan H. SteinUniversity of Connecticut
![Page 26: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/26.jpg)
Random Variables
Random variables are numerical values associated with outcomesof experiments.
The letter X is the most common symbol used to represent arandom variable.
If we consider a straight bet of $1 to be an experiment and X tobe the net winnings for the player, then X can take on either thevalue of 35 (if the player wins) or −1 (if the player loses).
If we denote the probability X takes on a certain value k by
P(X = k),
then P(X = 35) =1
38and P(X = −1) =
37
38.
Alan H. SteinUniversity of Connecticut
![Page 27: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/27.jpg)
Random Variables
Random variables are numerical values associated with outcomesof experiments.
The letter X is the most common symbol used to represent arandom variable.
If we consider a straight bet of $1 to be an experiment and X tobe the net winnings for the player, then X can take on either thevalue of 35 (if the player wins) or −1 (if the player loses).
If we denote the probability X takes on a certain value k by
P(X = k), then P(X = 35) =1
38and P(X = −1) =
37
38.
Alan H. SteinUniversity of Connecticut
![Page 28: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/28.jpg)
Mathematical Expectation
Mathematical expectation is essentially what we can expect theaverage value of a random variable to be close to.
We denote it byE (X ).
In this case, we expect E (X ) = − 1
19, since we expect to lose an
average of $1
19per bet.
Alan H. SteinUniversity of Connecticut
![Page 29: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/29.jpg)
Mathematical Expectation
Mathematical expectation is essentially what we can expect theaverage value of a random variable to be close to. We denote it byE (X ).
In this case, we expect E (X ) = − 1
19, since we expect to lose an
average of $1
19per bet.
Alan H. SteinUniversity of Connecticut
![Page 30: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/30.jpg)
Mathematical Expectation
Mathematical expectation is essentially what we can expect theaverage value of a random variable to be close to. We denote it byE (X ).
In this case, we expect E (X ) = − 1
19,
since we expect to lose an
average of $1
19per bet.
Alan H. SteinUniversity of Connecticut
![Page 31: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/31.jpg)
Mathematical Expectation
Mathematical expectation is essentially what we can expect theaverage value of a random variable to be close to. We denote it byE (X ).
In this case, we expect E (X ) = − 1
19, since we expect to lose an
average of $1
19per bet.
Alan H. SteinUniversity of Connecticut
![Page 32: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/32.jpg)
Calculating Mathematical Expectation
If we look at our earlier calculation, we can infer a reasonabledefinition of mathematical expectation.
Imagining playing 38 times, we took the 1 expected win, with Xbeing 35 that time, the 37 expected losses, with X being −1 thosetimes, and took 35− 37 · 1 = −2. We then divided by 38 to get−2
38= − 1
19.
If we unravel the calculation, we can write
− 1
19=
−2
38=
35− 37 · 138
=35 · 1− 1 · 37
38=
35 · 1 + (−1) · 37
38=
35 · 1
38+ (−1) · 37
38= 35P(X = 35) + (−1)P(X = −1).
Alan H. SteinUniversity of Connecticut
![Page 33: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/33.jpg)
Calculating Mathematical Expectation
If we look at our earlier calculation, we can infer a reasonabledefinition of mathematical expectation.
Imagining playing 38 times, we took the 1 expected win, with Xbeing 35 that time, the 37 expected losses, with X being −1 thosetimes, and took
35− 37 · 1 = −2. We then divided by 38 to get−2
38= − 1
19.
If we unravel the calculation, we can write
− 1
19=
−2
38=
35− 37 · 138
=35 · 1− 1 · 37
38=
35 · 1 + (−1) · 37
38=
35 · 1
38+ (−1) · 37
38= 35P(X = 35) + (−1)P(X = −1).
Alan H. SteinUniversity of Connecticut
![Page 34: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/34.jpg)
Calculating Mathematical Expectation
If we look at our earlier calculation, we can infer a reasonabledefinition of mathematical expectation.
Imagining playing 38 times, we took the 1 expected win, with Xbeing 35 that time, the 37 expected losses, with X being −1 thosetimes, and took 35− 37 · 1 = −2.
We then divided by 38 to get−2
38= − 1
19.
If we unravel the calculation, we can write
− 1
19=
−2
38=
35− 37 · 138
=35 · 1− 1 · 37
38=
35 · 1 + (−1) · 37
38=
35 · 1
38+ (−1) · 37
38= 35P(X = 35) + (−1)P(X = −1).
Alan H. SteinUniversity of Connecticut
![Page 35: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/35.jpg)
Calculating Mathematical Expectation
If we look at our earlier calculation, we can infer a reasonabledefinition of mathematical expectation.
Imagining playing 38 times, we took the 1 expected win, with Xbeing 35 that time, the 37 expected losses, with X being −1 thosetimes, and took 35− 37 · 1 = −2. We then divided by 38 to get−2
38= − 1
19.
If we unravel the calculation, we can write
− 1
19=
−2
38=
35− 37 · 138
=35 · 1− 1 · 37
38=
35 · 1 + (−1) · 37
38=
35 · 1
38+ (−1) · 37
38= 35P(X = 35) + (−1)P(X = −1).
Alan H. SteinUniversity of Connecticut
![Page 36: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/36.jpg)
Calculating Mathematical Expectation
If we look at our earlier calculation, we can infer a reasonabledefinition of mathematical expectation.
Imagining playing 38 times, we took the 1 expected win, with Xbeing 35 that time, the 37 expected losses, with X being −1 thosetimes, and took 35− 37 · 1 = −2. We then divided by 38 to get−2
38= − 1
19.
If we unravel the calculation, we can write
− 1
19=
−2
38=
35− 37 · 138
=35 · 1− 1 · 37
38=
35 · 1 + (−1) · 37
38=
35 · 1
38+ (−1) · 37
38= 35P(X = 35) + (−1)P(X = −1).
Alan H. SteinUniversity of Connecticut
![Page 37: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/37.jpg)
Calculating Mathematical Expectation
If we look at our earlier calculation, we can infer a reasonabledefinition of mathematical expectation.
Imagining playing 38 times, we took the 1 expected win, with Xbeing 35 that time, the 37 expected losses, with X being −1 thosetimes, and took 35− 37 · 1 = −2. We then divided by 38 to get−2
38= − 1
19.
If we unravel the calculation, we can write
− 1
19=
−2
38=
35− 37 · 138
=35 · 1− 1 · 37
38=
35 · 1 + (−1) · 37
38=
35 · 1
38+ (−1) · 37
38= 35P(X = 35) + (−1)P(X = −1).
Alan H. SteinUniversity of Connecticut
![Page 38: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/38.jpg)
Calculating Mathematical Expectation
If we look at our earlier calculation, we can infer a reasonabledefinition of mathematical expectation.
Imagining playing 38 times, we took the 1 expected win, with Xbeing 35 that time, the 37 expected losses, with X being −1 thosetimes, and took 35− 37 · 1 = −2. We then divided by 38 to get−2
38= − 1
19.
If we unravel the calculation, we can write
− 1
19=
−2
38=
35− 37 · 138
=
35 · 1− 1 · 37
38=
35 · 1 + (−1) · 37
38=
35 · 1
38+ (−1) · 37
38= 35P(X = 35) + (−1)P(X = −1).
Alan H. SteinUniversity of Connecticut
![Page 39: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/39.jpg)
Calculating Mathematical Expectation
If we look at our earlier calculation, we can infer a reasonabledefinition of mathematical expectation.
Imagining playing 38 times, we took the 1 expected win, with Xbeing 35 that time, the 37 expected losses, with X being −1 thosetimes, and took 35− 37 · 1 = −2. We then divided by 38 to get−2
38= − 1
19.
If we unravel the calculation, we can write
− 1
19=
−2
38=
35− 37 · 138
=35 · 1− 1 · 37
38=
35 · 1 + (−1) · 37
38=
35 · 1
38+ (−1) · 37
38= 35P(X = 35) + (−1)P(X = −1).
Alan H. SteinUniversity of Connecticut
![Page 40: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/40.jpg)
Calculating Mathematical Expectation
If we look at our earlier calculation, we can infer a reasonabledefinition of mathematical expectation.
Imagining playing 38 times, we took the 1 expected win, with Xbeing 35 that time, the 37 expected losses, with X being −1 thosetimes, and took 35− 37 · 1 = −2. We then divided by 38 to get−2
38= − 1
19.
If we unravel the calculation, we can write
− 1
19=
−2
38=
35− 37 · 138
=35 · 1− 1 · 37
38=
35 · 1 + (−1) · 37
38=
35 · 1
38+ (−1) · 37
38= 35P(X = 35) + (−1)P(X = −1).
Alan H. SteinUniversity of Connecticut
![Page 41: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/41.jpg)
Calculating Mathematical Expectation
If we look at our earlier calculation, we can infer a reasonabledefinition of mathematical expectation.
Imagining playing 38 times, we took the 1 expected win, with Xbeing 35 that time, the 37 expected losses, with X being −1 thosetimes, and took 35− 37 · 1 = −2. We then divided by 38 to get−2
38= − 1
19.
If we unravel the calculation, we can write
− 1
19=
−2
38=
35− 37 · 138
=35 · 1− 1 · 37
38=
35 · 1 + (−1) · 37
38=
35 · 1
38+ (−1) · 37
38=
35P(X = 35) + (−1)P(X = −1).
Alan H. SteinUniversity of Connecticut
![Page 42: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/42.jpg)
Calculating Mathematical Expectation
If we look at our earlier calculation, we can infer a reasonabledefinition of mathematical expectation.
Imagining playing 38 times, we took the 1 expected win, with Xbeing 35 that time, the 37 expected losses, with X being −1 thosetimes, and took 35− 37 · 1 = −2. We then divided by 38 to get−2
38= − 1
19.
If we unravel the calculation, we can write
− 1
19=
−2
38=
35− 37 · 138
=35 · 1− 1 · 37
38=
35 · 1 + (−1) · 37
38=
35 · 1
38+ (−1) · 37
38= 35P(X = 35) + (−1)P(X = −1).
Alan H. SteinUniversity of Connecticut
![Page 43: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/43.jpg)
Abstracting to Get a Formula
The calculation E (X ) = 35P(X = 35) + (−1)P(X = −1) suggestswe calculate mathematical expectation by taking each possiblevalue of the random variable, multiply it by the probability therandom variable takes on that value, and add the productstogether.
Symbolically, we may write this as
E (X ) =∑
kP(X = k),where the sum is taken over all the possible values for X .
Alan H. SteinUniversity of Connecticut
![Page 44: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/44.jpg)
Abstracting to Get a Formula
The calculation E (X ) = 35P(X = 35) + (−1)P(X = −1) suggestswe calculate mathematical expectation by taking each possiblevalue of the random variable, multiply it by the probability therandom variable takes on that value, and add the productstogether.
Symbolically, we may write this as
E (X ) =∑
kP(X = k),where the sum is taken over all the possible values for X .
Alan H. SteinUniversity of Connecticut
![Page 45: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/45.jpg)
Abstracting to Get a Formula
The calculation E (X ) = 35P(X = 35) + (−1)P(X = −1) suggestswe calculate mathematical expectation by taking each possiblevalue of the random variable, multiply it by the probability therandom variable takes on that value, and add the productstogether.
Symbolically, we may write this as
E (X ) =∑
kP(X = k),where the sum is taken over all the possible values for X .
Alan H. SteinUniversity of Connecticut
![Page 46: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/46.jpg)
Probability Spaces
Recall:
We consider experiments with a finite set of possible outcomes.
We call the set of possible outcomes the sample space and theindividual outcomes are called sample points.
We’re interested in the likelihood of each possibility and assign,hopefully in a meaningful way, a probability to each.
Definition (Probability)
A probability is a number between 0 and 1.
Definition (Probability Space)
A probability space is a sample space S for which a probabilityp(x) has been assigned to each sample point x such that the sum∑
x∈S p(x) of the probabilities is 1.
Alan H. SteinUniversity of Connecticut
![Page 47: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/47.jpg)
Probability Spaces
Recall:
We consider experiments with a finite set of possible outcomes.
We call the set of possible outcomes the sample space and theindividual outcomes are called sample points.
We’re interested in the likelihood of each possibility and assign,hopefully in a meaningful way, a probability to each.
Definition (Probability)
A probability is a number between 0 and 1.
Definition (Probability Space)
A probability space is a sample space S for which a probabilityp(x) has been assigned to each sample point x such that the sum∑
x∈S p(x) of the probabilities is 1.
Alan H. SteinUniversity of Connecticut
![Page 48: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/48.jpg)
Probability Spaces
Recall:
We consider experiments with a finite set of possible outcomes.
We call the set of possible outcomes the sample space
and theindividual outcomes are called sample points.
We’re interested in the likelihood of each possibility and assign,hopefully in a meaningful way, a probability to each.
Definition (Probability)
A probability is a number between 0 and 1.
Definition (Probability Space)
A probability space is a sample space S for which a probabilityp(x) has been assigned to each sample point x such that the sum∑
x∈S p(x) of the probabilities is 1.
Alan H. SteinUniversity of Connecticut
![Page 49: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/49.jpg)
Probability Spaces
Recall:
We consider experiments with a finite set of possible outcomes.
We call the set of possible outcomes the sample space and theindividual outcomes are called sample points.
We’re interested in the likelihood of each possibility and assign,hopefully in a meaningful way, a probability to each.
Definition (Probability)
A probability is a number between 0 and 1.
Definition (Probability Space)
A probability space is a sample space S for which a probabilityp(x) has been assigned to each sample point x such that the sum∑
x∈S p(x) of the probabilities is 1.
Alan H. SteinUniversity of Connecticut
![Page 50: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/50.jpg)
Probability Spaces
Recall:
We consider experiments with a finite set of possible outcomes.
We call the set of possible outcomes the sample space and theindividual outcomes are called sample points.
We’re interested in the likelihood of each possibility and assign,hopefully in a meaningful way, a probability to each.
Definition (Probability)
A probability is a number between 0 and 1.
Definition (Probability Space)
A probability space is a sample space S for which a probabilityp(x) has been assigned to each sample point x such that the sum∑
x∈S p(x) of the probabilities is 1.
Alan H. SteinUniversity of Connecticut
![Page 51: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/51.jpg)
Probability Spaces
Recall:
We consider experiments with a finite set of possible outcomes.
We call the set of possible outcomes the sample space and theindividual outcomes are called sample points.
We’re interested in the likelihood of each possibility and assign,hopefully in a meaningful way, a probability to each.
Definition (Probability)
A probability is a number between 0 and 1.
Definition (Probability Space)
A probability space is a sample space S for which a probabilityp(x) has been assigned to each sample point x such that the sum∑
x∈S p(x) of the probabilities is 1.
Alan H. SteinUniversity of Connecticut
![Page 52: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/52.jpg)
Probability Spaces
Recall:
We consider experiments with a finite set of possible outcomes.
We call the set of possible outcomes the sample space and theindividual outcomes are called sample points.
We’re interested in the likelihood of each possibility and assign,hopefully in a meaningful way, a probability to each.
Definition (Probability)
A probability is a number between 0 and 1.
Definition (Probability Space)
A probability space is a sample space S for which a probabilityp(x) has been assigned to each sample point x such that the sum∑
x∈S p(x) of the probabilities is 1.
Alan H. SteinUniversity of Connecticut
![Page 53: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/53.jpg)
Examples
Consider the sample space {H,T}, with probabilities
p(H) = p(T ) =1
2.
This may be considered a model for flipping a
coin a single time.
Consider the sample space {H0,H1,H2}, with probabilities
p(H0) =1
4, p(H1) =
1
2, p(H2) =
1
4. This may be considered a
model for flipping a coin twice.
The first example is known as an Equiprobable Space; the secondis not an equiprobable space.
Alan H. SteinUniversity of Connecticut
![Page 54: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/54.jpg)
Examples
Consider the sample space {H,T}, with probabilities
p(H) = p(T ) =1
2. This may be considered a model for flipping a
coin a single time.
Consider the sample space {H0,H1,H2}, with probabilities
p(H0) =1
4, p(H1) =
1
2, p(H2) =
1
4. This may be considered a
model for flipping a coin twice.
The first example is known as an Equiprobable Space; the secondis not an equiprobable space.
Alan H. SteinUniversity of Connecticut
![Page 55: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/55.jpg)
Examples
Consider the sample space {H,T}, with probabilities
p(H) = p(T ) =1
2. This may be considered a model for flipping a
coin a single time.
Consider the sample space {H0,H1,H2}, with probabilities
p(H0) =1
4, p(H1) =
1
2, p(H2) =
1
4.
This may be considered a
model for flipping a coin twice.
The first example is known as an Equiprobable Space; the secondis not an equiprobable space.
Alan H. SteinUniversity of Connecticut
![Page 56: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/56.jpg)
Examples
Consider the sample space {H,T}, with probabilities
p(H) = p(T ) =1
2. This may be considered a model for flipping a
coin a single time.
Consider the sample space {H0,H1,H2}, with probabilities
p(H0) =1
4, p(H1) =
1
2, p(H2) =
1
4. This may be considered a
model for flipping a coin twice.
The first example is known as an Equiprobable Space; the secondis not an equiprobable space.
Alan H. SteinUniversity of Connecticut
![Page 57: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/57.jpg)
Examples
Consider the sample space {H,T}, with probabilities
p(H) = p(T ) =1
2. This may be considered a model for flipping a
coin a single time.
Consider the sample space {H0,H1,H2}, with probabilities
p(H0) =1
4, p(H1) =
1
2, p(H2) =
1
4. This may be considered a
model for flipping a coin twice.
The first example is known as an Equiprobable Space; the secondis not an equiprobable space.
Alan H. SteinUniversity of Connecticut
![Page 58: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/58.jpg)
Equiprobable Spaces
Definition (Equiprobable Space)
A probability space P is called an equiprobable space if eachsample point has the same probability.
If an equiprobabe space P has n points, we’ll write |P| = n, then
p(x) =1
n∀x ∈ P.
In an equiprobable space, if we have an event E ⊂ P, then
p(E ) =|E |n
.
It’s obviously important to be able to count the number of samplepoints comprising events. The science of counting the size of a setis called combinatorics.
Alan H. SteinUniversity of Connecticut
![Page 59: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/59.jpg)
Equiprobable Spaces
Definition (Equiprobable Space)
A probability space P is called an equiprobable space if eachsample point has the same probability.
If an equiprobabe space P has n points,
we’ll write |P| = n, then
p(x) =1
n∀x ∈ P.
In an equiprobable space, if we have an event E ⊂ P, then
p(E ) =|E |n
.
It’s obviously important to be able to count the number of samplepoints comprising events. The science of counting the size of a setis called combinatorics.
Alan H. SteinUniversity of Connecticut
![Page 60: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/60.jpg)
Equiprobable Spaces
Definition (Equiprobable Space)
A probability space P is called an equiprobable space if eachsample point has the same probability.
If an equiprobabe space P has n points, we’ll write |P| = n,
then
p(x) =1
n∀x ∈ P.
In an equiprobable space, if we have an event E ⊂ P, then
p(E ) =|E |n
.
It’s obviously important to be able to count the number of samplepoints comprising events. The science of counting the size of a setis called combinatorics.
Alan H. SteinUniversity of Connecticut
![Page 61: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/61.jpg)
Equiprobable Spaces
Definition (Equiprobable Space)
A probability space P is called an equiprobable space if eachsample point has the same probability.
If an equiprobabe space P has n points, we’ll write |P| = n, then
p(x) =1
n∀x ∈ P.
In an equiprobable space, if we have an event E ⊂ P, then
p(E ) =|E |n
.
It’s obviously important to be able to count the number of samplepoints comprising events. The science of counting the size of a setis called combinatorics.
Alan H. SteinUniversity of Connecticut
![Page 62: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/62.jpg)
Equiprobable Spaces
Definition (Equiprobable Space)
A probability space P is called an equiprobable space if eachsample point has the same probability.
If an equiprobabe space P has n points, we’ll write |P| = n, then
p(x) =1
n∀x ∈ P.
In an equiprobable space, if we have an event E ⊂ P, then
p(E ) =|E |n
.
It’s obviously important to be able to count the number of samplepoints comprising events. The science of counting the size of a setis called combinatorics.
Alan H. SteinUniversity of Connecticut
![Page 63: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/63.jpg)
Equiprobable Spaces
Definition (Equiprobable Space)
A probability space P is called an equiprobable space if eachsample point has the same probability.
If an equiprobabe space P has n points, we’ll write |P| = n, then
p(x) =1
n∀x ∈ P.
In an equiprobable space, if we have an event E ⊂ P, then
p(E ) =|E |n
.
It’s obviously important to be able to count the number of samplepoints comprising events.
The science of counting the size of a setis called combinatorics.
Alan H. SteinUniversity of Connecticut
![Page 64: roulette - math.uconn.edustein/math1070/Slides/math1070-150... · A roulette wheel contains 38 slots, numbered 0, 00, and 1,2,3,...36. When the wheel is spun, a ball eventually falls](https://reader031.fdocuments.in/reader031/viewer/2022022801/5c717c8509d3f2282f8c2e33/html5/thumbnails/64.jpg)
Equiprobable Spaces
Definition (Equiprobable Space)
A probability space P is called an equiprobable space if eachsample point has the same probability.
If an equiprobabe space P has n points, we’ll write |P| = n, then
p(x) =1
n∀x ∈ P.
In an equiprobable space, if we have an event E ⊂ P, then
p(E ) =|E |n
.
It’s obviously important to be able to count the number of samplepoints comprising events. The science of counting the size of a setis called combinatorics.
Alan H. SteinUniversity of Connecticut